Problem 88 - Intermediate Algebra

Question

Aluminum Foil. Find the number of square feet of aluminum foil on a roll if the dimensions printed on the box are \(8 \frac{1}{3}\) yards \(\times 12\) inches.

Step-by-Step Solution

Answer

The roll of aluminum foil contains 25 square feet.

Step 1: Convert Yards to Feet

The length of the aluminum foil is given in yards. Convert this to feet since there are 3 feet in a yard. \[ 8 \frac{1}{3} \text{ yards} = \left(8 + \frac{1}{3}\right) \times 3 = \frac{25}{3} \times 3 = 25 \text{ feet} \]

Step 2: Convert Inches to Feet

The width of the aluminum foil is given in inches. Convert this measurement to feet, noting that there are 12 inches in a foot.\[ 12 \text{ inches} = \frac{12}{12} \text{ feet} = 1 \text{ foot} \]

Step 3: Calculate the Area in Square Feet

Now that both dimensions are in feet, multiply them to find the area in square feet. \[ \text{Area} = 25 \text{ feet} \times 1 \text{ foot} = 25 \text{ square feet} \]

Key Concepts

Understanding Area Calculation

Area calculation is a fundamental aspect of geometry that helps us understand the size of a surface. When calculating the area, we're essentially finding out how many square units cover a specific surface.

For rectangular shapes, like the aluminum foil roll in the example, the area is straightforward to calculate. You simply multiply its length by its width. This works because you're determining how many square units—like squares with sides of 1 foot—fit within the shape. In our case:

• The length was found to be 25 feet.
• The width was 1 foot.
• By multiplying, we obtain: 25 feet × 1 foot = 25 square feet.

Thus, the total area of the aluminum foil is 25 square feet, indicating that it would cover an area equivalent to a square that is 25 feet on each side.

Calculating area is not only important in academics but also in various real-life situations, such as determining the amount of paint needed for a wall or the size of carpet required for a room.

Exploring Algebraic Problem Solving

Algebraic problem-solving is all about using mathematical relationships and operations to find unknown values. It often involves converting and manipulating equations.

In this exercise, we applied algebraic skills when converting measurements and calculating the area. The initial measurement was in yards, and we used algebraic manipulation to handle the fraction to convert into feet. The conversion steps involved:

• Understanding that 1 yard equals 3 feet.
• Breaking down the mixed number: convert 8 \(\frac{1}{3}\) yards to \(\frac{25}{3}\) yards for easy calculation.
• Multiplying by 3 to find the feet equivalent, thus confirming the 25 feet.

Algebra aids in making such conversions and calculations seamless by ensuring each step follows logically from previous ones, leading to solving for the area efficiently.

Navigating Measurement Units

Measurement units can differ, and effective problem-solving often requires converting between these to accomplish the task accurately. The units of measurement provide a common language to describe the dimensions of objects.

In our example, we dealt with both yards and inches, converting all measurements to feet.

• Yard to feet: Since 1 yard equals 3 feet, it's crucial to remember this when dealing with similar conversions.
• Inch to feet: As 12 inches make up 1 foot, it's equally important to apply this conversion when necessary.

When converting, focus on the relationships between units. This not only aids in calculating area but also ensures universal understanding across various applications.

By comprehending unit conversion, you can approach complex problems with confidence, knowing how to navigate between different units seamlessly to find your solution.